The aim of the investigation contained in this paper is to recover the full generality avoiding any extra hypothesis on the admissible free-crystal regions. This is achieved by retrieving compactness with respect to the free-crystal regions at least for any sequence of \(m\) -minimizers \((A_m,u_m)\in \mathcal {C}_m\) , and by combining the strategies of [1]} and [2]}.
More precisely, the use in [2]} of the Golab-type Theorem [4]} is avoided for the compactness of the free-crystal regions by adapting to our setting the classical density-estimate arguments first introduced for surface energies and the Mumford-Shah functional (see, e.g., [5]}, [6]}, [7]}), and then extended to the Griffith functional [8]}, [9]}.
